Undergrad: If you can do research now it helps set the groundwork for both medical school and residency applications. You have more time and leeway to learn how to do things. Do well on the MCAT and get in to the best/cheapest medical school you can. A medical school with a neurosurgery program, though not imperative, is more desirable than one without because it can be your launch pad to getting a residency.
Medical School: Do well in preclinical classes. If you have grades/honors, try to do well. Step 1 is the great equalizer, do the best you can. Third year rotations are again important, notably surgery. While a letter from the surgery chairman is not necessary, having honors in surgery looks good. Most applicants have some sort of research project during medical school. These range from a case report to a clinical study. If you know you want to do neurosurgery early, get involved with something with the department as early as possible and push it through. Even a chart review for a retrospective cohort trial can be done over the summer between 1st/2nd year. This both helps establish that you are interested, teaches you about neurosurgery and the terms/procedures, and introduces you to the department. During the 4th year do sub-internships on the neurosurgery service at your home program and 1-3 away programs. In neurosurgery in particular these away rotations are important because the letter from the away program helps establish that you are a team player and a good fit. It's a small field and everybody knows everybody.
Residency: Each program structures it differently and it can vary widely, but in general the following applies.
Internship: Up to 6 months of neurosurgery. Some general surgery and electives (neurology, critical care, interventional neuroradiology).
PGY 2-6: A mix of junior resident in the unit, on the floor, consults. Depending on the model, some break the hospital in to services i.e. pediatrics, spine, cerebrovascular, trauma while others lump them together still others use an apprentice model where a resident is assigned an attending(s). For the most part juniors (PGY-2) do scut and seniors (PGY6) operate. Most programs have 1-2 years in the middle for electives and research time. This can either be protected from the call schedule, where you do not take call, or not where you are mixed in to the call schedule in varying ways (2 calls/month, weekends, etc.)
PGY 7: Chief year. Run the service/operate. Some programs have moved to having the chief be a junior faculty member with admitting privileges.
Call is very program dependent. Some are q3 for the junior years and others are q12.
Fellowship: Specialties include cerebrovascular, endovascular, spine, peripheral nerve, skull base/endoscopic, pediatrics, trauma, and functional. Depending on the structure, fellows get to pick cases over the chief, or are tasked to work with the chief and walk them through cases. Most take home call, sometimes they have admitting privileges.